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Theorem f1cocnv1 5668
Description: Composition of an injective function with its converse. (Contributed by FL, 11-Nov-2011.)
Assertion
Ref Expression
f1cocnv1  |-  ( F : A -1-1-> B  -> 
( `' F  o.  F )  =  (  _I  |`  A )
)

Proof of Theorem f1cocnv1
StepHypRef Expression
1 f1f1orn 5648 . 2  |-  ( F : A -1-1-> B  ->  F : A -1-1-onto-> ran  F )
2 f1ococnv1 5667 . 2  |-  ( F : A -1-1-onto-> ran  F  ->  ( `' F  o.  F
)  =  (  _I  |`  A ) )
31, 2syl 16 1  |-  ( F : A -1-1-> B  -> 
( `' F  o.  F )  =  (  _I  |`  A )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    _I cid 4457   `'ccnv 4840   ran crn 4842    |` cres 4843    o. ccom 4845   -1-1->wf1 5414   -1-1-onto->wf1o 5416
This theorem is referenced by:  f1eqcocnv  5991  domss2  7229  diophrw  26711
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424
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